The local behavior of principal and chordal principal cluster sets

Author:
John T. Gresser

Journal:
Trans. Amer. Math. Soc. **165** (1972), 323-332

MSC:
Primary 30A72

DOI:
https://doi.org/10.1090/S0002-9947-1972-0294643-3

MathSciNet review:
0294643

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Abstract: Let *K* be the unit circle, and let *f* be a function whose domain is the open unit disk and whose range is a subset of the Riemann sphere. We define a set, called the boundary principal cluster set of *f* at , which characterizes the behavior of the principal cluster sets of *f* at points which are near and distinct from . It is shown that if *f* is continuous, then the principal and boundary principal cluster sets of *f* at are equal for nearly every point . A similar result holds for chordal principal cluster sets. Examples are provided that indicate directions in which the result cannot be improved. Some results concerning points that are accessible through sets which are unions of arcs are also presented.

**[1]**F. Bagemihl, G. Piranian and G. S. Young,*Intersections of cluster sets*, Bul. Inst. Politehn. Iaşi (N.S.)**5**(9) (1959), no. 3-4, 29-34. MR**22**#8118. MR**0117337 (22:8118)****[2]**F. Bagemihl,*Some results and problems concerning chordal principal cluster sets*, Nagoya Math. J.**29**(1967), 7-18. MR**35**#4425. MR**0213565 (35:4425)****[3]**C. L. Belna,*Intersections of arc-cluster sets for meromorphic functions*, Nagoya Math. J.**40**(1970), 213-221. MR**0271357 (42:6240)****[4]**E. F. Collingwood and A. J. Lohwater,*The theory of cluster sets*, Cambridge Tracts in Math. and Math. Phys., no. 56, Cambridge Univ. Press, Cambridge, 1966. MR**38**#325. MR**0231999 (38:325)****[5]**P. Lappan and D. C. Rung,*Normal functions and non-tangential boundary arcs*, Canad. J. Math.**18**(1966), 256-264. MR**33**#7553. MR**0199407 (33:7553)****[6]**K. Noshiro,*Cluster sets*, Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Heft 28, Springer-Verlag, Berlin, 1960. MR**24**#A3295. MR**0133464 (24:A3295)****[7]**G. T. Whyburn,*Analytic topology*, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R. I., 1963. MR**32**#425. MR**0182943 (32:425)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0294643-3

Keywords:
Principal cluster set,
chordal principal cluster set,
accessible points,
Baire category,
three-arc property

Article copyright:
© Copyright 1972
American Mathematical Society